Joe:
This problem is nearly always related to one of three things:
1) The water line itself is cracked or pinholed (happens over time when you pull the refrigerator out for cleaning).
2) The compression fittings are somehow damaged.
3) The supply tubing flares are somehow damaged.
The fix is usually very simple - buy an icemaker installation kit and replace everything. The kit has everything you usually need. Here's what to do:
Pull out the refrigerator and follow the icemaker water supply tubing (might be plastic, might be copper) to where it goes into a brass fitting on the back of the fridge. That fitting unscrews with a wrench (WARNING: the piece the fitting screws to can be broken, so be sure and hold on to it with one hand while unscrewing the fitting).
Once the fitting is unscrewed, you will be able to slide it back along the length of tubing, exposing the end of the tube. If the tubing is copper, the end will be flared. If it is plastic, there will be a little brass piece that is flared inserted into the end of the tubing. There will also be a little brass ring around the end of the tubing, on the outside of the tubing. This is called a compression fitting.
What generally cures your problem is to replace all these pieces, following the instructions on the icemaker installation kit. Every hardware store carries them, in both plastic and copper. Both plastic and copper have their fans, and I'm not getting into that debate.
It is really a pretty easy repair, just a little tedious. You will also have to find where your icemaker supply tubing connects to your cold water line; it's called a supply tee. You should shut off the old tee's little shutoff valve where it connects to the cold water pipe, and cut off the old tubing. Then install a new tee, connecting up your new tubing to that. Again, complete instructions are on the icemaker installation kit.
If this does not cure the problem, there are two other possibilities. Either you did not tighten things correctly when you put in the new icemaker installation kit (you can overtighten as well as fail to tighten enough), or you have a cracked part somewhere on the refrigerator, in which case you should get back to us.
Try the icemaker kit first; they're cheap, usually cure the problem, and if you have a cracked part on the fridge, you'd end up having to replace it anyway. Here's a link to installation instructions for the kit (they're all pretty much the same):